lim┬(x→∞)〖(1/x+2^(1/x) )^x 〗求极限,
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![lim┬(x→∞)〖(1/x+2^(1/x) )^x 〗求极限,](/uploads/image/z/6890311-55-1.jpg?t=lim%E2%94%AC%28x%E2%86%92%E2%88%9E%29%26%238289%3B%E3%80%96%281%2Fx%2B2%5E%281%2Fx%29+%29%5Ex+%E3%80%97%E6%B1%82%E6%9E%81%E9%99%90%2C)
lim┬(x→∞)〖(1/x+2^(1/x) )^x 〗求极限,
lim┬(x→∞)〖(1/x+2^(1/x) )^x 〗求极限,
lim┬(x→∞)〖(1/x+2^(1/x) )^x 〗求极限,
lim(x→∞) [1/x +2^(1/x)] ^x 令 t=1/x
原式= lim(t->0) (t + 2^t) ^ (1/t)
= e^ lim(t->0) ln( t+2^t) / t
lim(t->0) ln(t+2^t) / t
= lim(t->0) ln(1+ t + 2^t -1) / t 等价无穷小代换 u = t+2^t -1 -> 0,ln(1+u) u
= lim(t->0) ( t+ 2^t -1) / t 等价无穷小代换 2^t - 1 t * ln2
= lim(t->0) ( t+ ln2* t) / t = 1 + ln2
原式 = e^(1+ln2) = 2e